A321662 Number of non-isomorphic multiset partitions of weight n whose incidence matrix has all distinct entries.

1, 1, 1, 3, 3, 5, 13, 15, 23, 33, 49, 59, 83, 101, 133, 281, 321, 477, 655, 941, 1249, 1795, 2241, 3039, 3867, 5047, 6257, 8063, 11459, 13891, 18165, 23149, 29975, 37885, 49197, 61829, 89877, 109165, 145673, 185671, 246131, 310325, 408799, 514485, 668017, 871383

Offset

0,4

Comments

The incidence matrix of a multiset partition has entry (i, j) equal to the multiplicity of vertex i in part j.

Also the number of positive integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with all different entries.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{k>=1} (A121860(k) + A121860(k+1) - 2)*A008289(n,k) for n > 0. - Andrew Howroyd, Nov 17 2018

Example

Non-isomorphic representatives of the a(3) = 3 through a(7) = 15 multiset partitions:
  {{111}}    {{1111}}    {{11111}}    {{111111}}      {{1111111}}
  {{122}}    {{1222}}    {{11222}}    {{112222}}      {{1112222}}
  {{1}{11}}  {{1}{111}}  {{12222}}    {{122222}}      {{1122222}}
                         {{1}{1111}}  {{122333}}      {{1222222}}
                         {{11}{111}}  {{1}{11111}}    {{1223333}}
                                      {{11}{1111}}    {{1}{111111}}
                                      {{1}{11222}}    {{11}{11111}}
                                      {{11}{1222}}    {{111}{1111}}
                                      {{112}{222}}    {{1}{112222}}
                                      {{122}{222}}    {{11}{12222}}
                                      {{2}{11222}}    {{112}{2222}}
                                      {{22}{1222}}    {{122}{2222}}
                                      {{1}{11}{111}}  {{2}{112222}}
                                                      {{22}{12222}}
                                                      {{1}{11}{1111}}

Mathematica

(* b = A121860 *) b[n_] := Sum[n!/(d! (n/d)!), {d, Divisors[n]}];
(* c = A008289 *) c[n_, k_] := c[n, k] = If[n < k || k < 1, 0, If[n == 1, 1, c[n - k, k] + c[n - k, k - 1]]];
a[n_] := If[n == 0, 1, Sum[ (b[k] + b[k + 1] - 2) c[n, k], {k, 1, n}]];
a /@ Range[0, 45] (* Jean-François Alcover, Sep 14 2019 *)

Prog

(PARI) \\ here b(n) is A121860(n).
b(n)={sumdiv(n, d, n!/(d!*(n/d)!))}
seq(n)={my(B=vector((sqrtint(8*(n+1))+1)\2, n, if(n==1, 1, b(n-1)+b(n)-2))); apply(p->sum(i=0, poldegree(p), B[i+1]*polcoef(p, i)), Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Nov 16 2018

Crossrefs

Cf. A000219, A007716, A008289, A059201, A114736, A117433, A120733, A121860, A321653, A321659, A321660, A321661.

Sequence in context: A079439 A231895 A218426 * A320176 A298478 A144419

Adjacent sequences: A321659 A321660 A321661 * A321663 A321664 A321665

Keyword

nonn

Author

Gus Wiseman, Nov 15 2018

Extensions

Terms a(11) and beyond from Andrew Howroyd, Nov 16 2018

Status

approved